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International Journal for Uncertainty Quantification

Impact factor: 1.000

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2011003089
pages 257-278

PINK NOISE, 1/fα NOISE, AND THEIR EFFECT ON SOLUTIONS OF DIFFERENTIAL EQUATIONS

Miroslav Stoyanov
Applied Mathematics Group, Computer Science and Mathematics Division, Oak Ridge National Laboratory, 1 Bethel Valley Road, P.O. Box 2008, Oak Ridge TN 37831-6164
Max Gunzburger
Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306-4120
John Burkardt
Department of Scientific Computing, Florida State University, Tallahassee, Florida 32306-4120, USA

ABSTRACT

White noise is a very common way to account for randomness in the inputs to partial differential equations, especially in cases where little is know about those inputs. On the other hand, pink noise, or more generally, colored noise having a power spectrum that decays as 1/fα, where f denotes the frequency and α Є (0; 2] has been found to accurately model many natural, social, economic, and other phenomena. Our goal in this paper is to study, in the context of simple linear and nonlinear two-point boundary-value problems, the effects of modeling random inputs as 1/fα random fields, including the white noise (α = 0), pink noise (α = 1), and brown noise (α = 2) cases. We show how such random fields can be approximated so that they can be used in computer simulations. We then show that the solutions of the differential equations exhibit a strong dependence on α, indicating that further examination of how randomness in partial differential equations is modeled and simulated is warranted.